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Q7 Define f(x)=³-3x+1, and o(a)=x²-2. The following information will be useful: f is irreducible over Q. • f(o(x)) = g(x)f(x) where g(x)=3-3x - 1. (You do not need to check either of these things although both are straightforward calculations.) (a) How many distinct roots does f have in C? Explain your answer. (b) Show that maps any root of f to another, distinct, root of f. I.e., if a is a root of f, then o(a) is a root of f, and moreover o(a) # a.

Q7 Define f(x)=³-3x+1, and o(a)=x²-2. The following information will be useful: f is irreducible over Q. • f(o(x)) = g(x)f(x) where g(x)=3-3x - 1. (You do not need to check either of these things although both are straightforward calculations.) (a) How many distinct roots does f have in C? Explain your answer. (b) Show that maps any root of f to another, distinct, root of f. I.e., if a is a root of f, then o(a) is a root of f, and moreover o(a) # a.

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter8: Polynomials
Section8.4: Zeros Of A Polynomial
Problem 31E
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Q7 Define f(x)=x²-3x+1, and (x)=x²-2. The following information will be useful: • f is irreducible over Q. f(g(x)) = g(x)f(x) where g(x) = 2³-3r - 1. (You do not need to check either of these things although both are straightforward calculations.) (a) How many distinct roots does f have in C? Explain your answer. (b) Show that maps any root of f to another, distinct, root of f. I.e., if a is a root of f, then o(a) is a root of f, and moreover o(a) # a. (c) Now let a be some root of f. Show that Q(a) must be the splitting field of f, and so Q(a): Q is normal. (d) What are the elements of the Galois group of f? What is this group isomorphic to? (e) How many intermediate fields does Q(a): Q have other than Q(a) and Q itself? Explain.
Transcribed Image Text:Q7 Define f(x)=x²-3x+1, and (x)=x²-2. The following information will be useful: • f is irreducible over Q. f(g(x)) = g(x)f(x) where g(x) = 2³-3r - 1. (You do not need to check either of these things although both are straightforward calculations.) (a) How many distinct roots does f have in C? Explain your answer. (b) Show that maps any root of f to another, distinct, root of f. I.e., if a is a root of f, then o(a) is a root of f, and moreover o(a) # a. (c) Now let a be some root of f. Show that Q(a) must be the splitting field of f, and so Q(a): Q is normal. (d) What are the elements of the Galois group of f? What is this group isomorphic to? (e) How many intermediate fields does Q(a): Q have other than Q(a) and Q itself? Explain.
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